You can interpret the line integral being zero to have some special meaning:
In physics, line integrals are used to calculate the (physical) work used to move an object (e.g. a hockey puck) along a path in some force field (e.g. the gravitational field).
A vector field in question which is a 2d-field ($F(x,y)= (x^2y,xy^2)$) might arise in Problems where the 3rd component doesn't matter, because we are moving the object on a flat table, for instance (and the form of $F$ comes from some electric field, and the puck is charged).
If we now move the object along a given path and the path integral is zero, then we didn't need to use any work to do it, i.e. we didn't need to work against the force field.
This is similar to the fact that lifting a hockey puck takes some effort, but it's really easy to move it on a flat, slippery surface.